Koya Shimokawa

Ochanomizu University · Department of Mathematics

With increasing ring-crossing number (c), knot theory predicts an exponential increase in the number of topologically different links of these interlocking structures, even for structures with the same ring number (n) and c. Here, we report the selective construction of two topologies of 12-crossing peptide [4]catenanes (n = 4, c = 12) from metal i...

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely many steps.

Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local" and, for several ring polymer models, we provide both theoretical and numerical evidence that, at equilibrium, the non-local configurations are more likely than the local ones. These characterisati...

We introduce the notions of Heegaard splittings and thin multiple Heegaard splittings of 1-submanifolds in compact orientable 3-manifolds, which are generalizations of those of bridge decompositions and thin positions. We show that ei-ther a thin multiple Heegaard splitting of 1-submanifold T is also a Heegaard splitting with minimal complexity or...

Which chiral knots can be unknotted in a single step by a + to − (+−) crossing change, and which by a − to + (−+) crossing change? Numerical results suggest that if a knot with 6 or fewer crossings can be unknotted by a +− crossing change then it cannot be unknotted by a −+ one, and vice versa. However, we exhibit one chiral 8-crossing knot and one...

This paper investigates the relationship between the signature and the crossing number of knots and links. We refine existing theorems and provide a comprehensive classification of links with specific properties, particularly those with signatures that deviate by a fixed amount from their crossing numbers. The main results include the identificatio...

Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-type K lattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition of K. This Knot Entr...

We employ a mathematical model to analyze stress chains in thermoplastic elastomers (TPEs) with a microphase-separated spherical structure composed of triblock copolymers. The model represents stress chains during uniaxial and biaxial extensions using networks of spherical domains connected by bridges. We advance previous research and discuss perma...

We show, for every positive integer $n$, there is an alternating knot having a boundary slope with denominator $n$. We make use of Kabaya's method for boundary slopes and the layered solid torus construction introducedby Jaco and Rubinstein and further developed by Howie et al.

We show, for every positive integer $n$, there is an alternating knot having a boundary slope with denominator $n$. We make use of Kabaya's method for boundary slopes and the layered solid torus construction introduced by Jaco and Rubinstein and further developed by Howie et al.

Knots and links are ubiquitous in chemical systems. Their structure can be responsible for a variety of physical and chemical properties, making them very important in materials development. In this article, we analyze the topological structures of interlocking molecules composed of metal-peptide rings using the concept of polyhedral links. To that...

We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss...

From polymer models, it has been conjectured that the exponential growth rate of the number of lattice polygons with knot-type $K$ is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot factor in the knot decomposition of $K$. Here we prove this conjecture for any knot or non-split link...

In this chapter, we will discuss the mathematical method used in analyses of topological polymers. First, we apply graph theory to define a notation for multi-cyclic polymers. We also consider the types of polymers and study the construction method. Second, we apply knot theory to multi-cyclic polymers. We analyze topological isomers derived from k...

We introduce the concept of a handlebody decomposition of a 3-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable 3-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topol...

Cavity creation is a key to the origin of biological functions. Small cavities such as enzyme pockets are created simply through liner peptide folding. Nature can create much larger cavities by threading and entangling large peptide rings, as learned from gigantic virus capsids, where not only chemical structures but the topology of threaded rings...

The control of topological chain properties is essential in biopolymer processes, including DNA transcription and replication promoted by topoisomerase enzymes. Cleavage and re-bonding of DNA chains enables transformation of the chain topologies between a trivial knot (simple ring) and higher knotted constructions, as well as linked counterparts[1,...

There are abundant examples in which the form of objects dictates their functions and properties at all dimensions and scales. In polymer chemistry and materials science, macromolecular structures have mostly been limited to linear or randomly branched forms. However, a variety of precisely controlled polymer topologies have been synthesized using...

As seen in Chapter 2, topology variety increases with increasing multicyclic graphs: e.g., dicyclic, tricyclic, tetracyclic. Multicyclic graphs are classified into spiro, bridge, fused, and hybrid types [1]. In this chapter, we present graph theory definitions of the various multicyclic graphs and characterize each type via construction and decompo...

In this chapter, we introduce polymers, long-chain molecules with diverse chemical compositions and structures. Topology can provide fundamental insights into the principle properties of polymers via their segment structures. We also present a brief description of the following chapters with respect to topological geometry and polymer chemistry.

In this chapter, we discuss topological isomers of multicyclic polymers by using knots, links, and spatial graphs. Essential references on knot theory are [1, 2, 3, 4]. Chemistry applications of knot theory and low-dimensional topology are widely discussed in [5, 6].

In this chapter, we describe the chemistry-based, hierarchical classification procedure in which a series of nonlinear, cyclic, and branched polymer architectures are classified from the molecular graph presentation of alkanes and cycloalkanes. We also discuss a systematic notation protocol for nonlinear polymer topologies, modified from the previo...

In this chapter, we introduce graph theory for analyzing structures of multicyclic polymers. An essential graph theory reference is [1].

In this monograph, we discussed and demonstrated ongoing developments in the unique collaboration of topological geometry and polymer chemistry. We described current topological polymer chemistry by highlighting the diverse nature of polymers with respect to both their chemistry and their line constructions. Topological analyses could provide funda...

We show that, for an alternating knot, the ratio of the diameter of the set of boundary slopes to the crossing number can be arbitrarily large.

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely many steps.

Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local" and, for several ring polymer models, we provide both theoretical and numerical evidence that, at equilibrium, the non-local configurations are more likely than the local ones. These characterisati...

In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif-FtsK can remove re...

In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif -FtsK can remove r...

Knots are ubiquitous in nature and their analysis has important implications in a wide variety of fields including fluid dynamics, material science and molecular and structural biology. In many systems particles are found in crowded environments hence it is natural to rigorously characterize the properties of knots in confined volumes. In this work...

We propose an algorithm for enumerating graphs representing polymer topologies. We also present experimental results on the algorithm. This is a preliminary report of the ongoing research.

For a compact connected 3-submanifold with connected boundary in the 3-sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a null-homologous link. As its corollary, we obtain a refinement of the Fox's re-embedding theorem.

The tangle method, first introduced by Ernst and Sumners in the late 1980s, uses tools from knot theory and low-dimensional topology to analyze the topological changes induced by site-specific recombination on a circular DNA substrate. Often, a recombination reaction can be modeled by a band surgery. Here we provide a brief description of the tangl...

Significance Newly replicated circular chromosomes are topologically linked. XerC/XerD- dif (XerCD- dif )–FtsK recombination acts in the replication termination region of the Escherichia coli chromosome to remove links introduced during homologous recombination and replication, whereas Topoisomerase IV removes replication links only. Based on gel m...

We characterize cutting arcs on fiber surfaces producing fiber surfaces. As corollaries we characterize band surgeries and generalized crossing changes between fibered links.

The protein recombinase can change the knot type of circular DNA. The action of a recombinase converting one knot into another knot is normally mathematically modeled by band surgery. Band surgeries on a 2-bridge knot N((4mn-1)/(2m)) yielding a (2,2k)-torus link are characterized. We apply this and other rational tangle surgery results to analyze X...

Volume confinement is a key determinant of the topology and geometry of a polymer. However, the direct relationship between the two is not fully understood. For instance, recent experimental studies have constructed P4 cosmids, i.e. P4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extract...

During site-specific recombination, the topology of circular DNA can change, e.g. unknotted molecules can become knotted or linked. We model Xer site-specific recombinations as the mathematical operation of band surgeries. In this paper, we consider band surgeries on knots with 7 and fewer crossings and links with 8 and fewer crossings.

We show that there exist infinitely many examples of pairs of knots, K_1 and K_2, that have no epimorphism $\pi_1(S^3\setminus K_1) \to \pi_1(S^3\setminus K_2)$ preserving peripheral structure although their A-polynomials have the factorization $A_{K_2}(L,M) \mid A_{K_1}(L,M)$. Our construction accounts for most of the known factorizations of this...

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numer...

We classify Dehn surgeries on (p,q,r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Agol and Lackenby's 6-theorem reduces the argument to knots with small indices p,q,r. We treat these using the Culler-Shalen norm of the SL(2,...

Let X be a norm curve in the SL(2,C)-character variety of a knot exterior M. Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two distinct non-zero classes a, b in H_1(\partial M, Z). We demonstrate that either X has exactly two associated strict boundary slopes \pm t, or else there are strict boundary slopes r_1 and r_2 with |r...

In this paper we discuss the relation between the combinatorial properties of cell decompositions of 3-spheres and the bridge index of knots contained in their 1-skeletons. The main result is to solve the conjecture of Ehrenborg and Hachimori which states that for a knot K in the 1-skeleton of a constructible 3-sphere satisfies e(K) ≥ 2b(K), where...

For some non-simple 2-component links in S3, we determine when a Dehn surgery yields S3.

We show that any Heegaard splitting of trivial arcs in a compression body is standard.

We show that any Heegaard splitting of the pair of the solid torus (@D2?S1) and its core loop (an interior point ?S1) is standard, using the notion of Heegaard splittings of pairs of 3-manifolds and properly imbedded graphs, which is defined in this paper.

In this article we show no Dehn surgery on nontrivial strongly invertible knots can yield the lens space L(2p; 1) for any integer p .I n order to do that, we determine band attaches to (2; 2p)-torus links producing the trivial knot.

We consider primeness, hyperbolicity, ∂-irreducibility and tangle sums of alternating tangles. We also study primeness and hyperbolicity of links and Dehn surgeries on knots admitting alternating tangle decompositions.

We show that any Heegaard splitting of the trivial knot in a compact orientable 3-manifold is standard.

We consider the parallelism of two strings in alternating tangles. We show that if there is a pair of parallel strings in an alternating tangle then its alternating diagrams satify certain conditions. As a corollary, for a knot admitting a decomposition into two alternating tangles with two or three strings, we prove that its non-trivial Dehn surge...

The cabling conjecture states that a non-trivial knot K in the 3-sphere is a cable knot or a torus knot if some Dehn surgery on K yields a reducible 3-manifold. We prove that symmetric knots satisfy this conjecture. (Gordon and Luecke also prove this independently ([GLu3]), by a method different from ours.)

In this article we consider tunnel number one alternating knots and links. We characterize tunnel number one alternating knots and links which have unknotting tunnels contained in regions of reduced alternating diagrams. We show that such a knot or link is a $2$-bridge or a Montesinos knot or link or the connected sum of the Hopf link and a $2$-bri...